Notes(1) the chorus is fragments of Cajun patois, with more or less phonetic spelling.
You will often find "Jockamo" rather than "Jockomo" for example
(and Andrew Katzenstein has suggested "Giacomo").
And you sometimes see "feeno ai nan" and "feenan."
I don't believe there is a "correct" spelling.(2) "chicken wire" is what it sounds as if Jerry is singing
(though on some other versions it sounds more like "chicko wiyo").
I haven't tracked this line down to any definitive "source" in other versions.
Reg Johnsey came up with this explanation:

The way country people celebrated Carnivale/Mardis
Gras was to make conical masks out of chicken wire and
decorate them, wearing them with costumes festooned
with strips of cloth.
So, the references to fixing someone's chicken wire
sounds like a joking threat to mess up their masks,
since part of the battle was how good the costumes
were.

(3) in some versions (eg 27 Feb 1990), Jerry sings
"My marraine see your marraine" for this line.
This is a line that Dr John sings as "My marraine see your parrain."
"Marraine" is French for "godmother,"
though in patois it is often used for "grandmother."
"Parrain" is godfather/grandfather.
(thanks to Adam Wasserman for the explanation)

For comparison, this is the version from 16 September 1990 (on Dick's Picks Vol 9):

On 14 June 1985, Bob Weir sang a couple of verses that are hard to decipher.
They seem to be lines from or a reference to another song "Meet The Boys On The Battlefront,"
recorded by Wild Tchoupitoulas (thanks to David Deephouse for this information):

Meet the boys on the battlefront
The Wild Tchoupitoulas gonna stomp some rump

Meet the boys on the battlefront, sing brother John is gone
Meet the boys on the battlefront, bye bye, cho cho

One additional verse that appears on some lyric collections is below.
I haven't yet confirmed whether it definitely sung by the Grateful Dead:

Well look at that girl all dressed in green
She hides a pistol where it can't be seen

Notes(a) issued as a bonus track in the box set Beyond Description (1973-1990)(b) this may be the version issued as a 7" vinyl single for
Record Store Day 2013, backed with recordings of the same song by Dr John and the
Dixie Chicks - but this is not yet confirmed.

Roots
According to Dr John in the liner notes to his 1972 album "Gumbo":

"The song was written and recorded back in the early 1950s by a New Orleans
singer named James Crawford who worked under the name of Sugar Boy & the
Cane Cutters. It was recorded in the 1960s by the Dixie Cups for Jerry
Leiber & Mike Stoller's Red Bird label, but the format we're following here
is Sugar Boy's original. Also in the group were Professor Longhair on piano,
Jake Myles, Big Boy Myles, Irv Bannister on guitar, and Eugene 'Bones' Jones
on drums. The group was also known as the Chipaka Shaweez. The song was
originally called 'Jockamo,' and it has a lot of Creole patois in it.
Jockamo means 'jester' in the old myth."

Thanks to Adam Wasserman for the following information:

Iko Iko (as well as other songs such as Big Chief, Hey Pokey-Way, New Suit,
Fire Water) has a very specific meaning.
They are all New Orleans Mardi Gras songs about the Black Indians.
Black Indians are parade crewes (tribes) that parade through the New Orleans streets
on Mardi Gras wearing extravagant ceremonial Indian clothes. They face off
when they meet and have battles of clothing, dancing, and singing.
The Spy Boy is a ceremonial position (the front runner who scouts out other
tribes to do battle with) as is the Flag Boy, Wild Man, and Big Chief.
Friends and family who follow are in the "second line" and are therefore
second liners.
So lines like "My spy boy to your spy boy, I'm gonna set your tail on fire"
are ceremonial challenges to the other tribe.

"Joc-a-mo-fee-no-ah-nah-nay, Joc-a-mo-fee-nah-nay"
is a ritual chant used by the Mardi Gras Indians
which has been around for so long the words are no longer clearly distinguishable,
and it has a well understood meaning of its own.
Very, very loosely translated it signifies "we mean business"
or "don't mess with us".
Originally it would have been Cajun
(a liberal mix of French and English)
and literally translates to "the fool we will not play today".

One additional comment on the origins/meaning of "Iko":

"Iko and un day are
Creole corruptions of the Gambian call ago! [pay attention] and the expected
response, which is amay! [I/we are listening].
Chuck Davis of the African-
American Dance Ensemble, which is based here in Durham, uses this device
ubiquitously when he acts as Griot (master storyteller/master of ceremonies).
When he calls "ago!" everyone is supposed to shout "amay!"--no matter what else
is going on. He likes to slip this into the middle of various narrations just
to make sure folks are paying attention. He also uses it as an introductory,
"calm down" sort of exercise before he starts to speak, or to quiet the crowd
if it gets noisy while he's speaking."

These are the lyrics from Sugar Boy Crawford's version
(thanks to Anita Cantor for help with the transcription):